List of top Mathematics Questions on Application of derivatives

If \[ f(x)=(1+x^3)(1+x^6)(1+x^{12})(1+x^{24}) \] then \(f'(-1)=\)
The domain of derivative of real valued function \[ f(x)=(x^2-x-2)|x^2+x-6| \] is
If the displacement of a particle at time $t$ ($0 < t < \pi$) is given by $s = 3 \sin 2t - 6 \cos t$, then the acceleration for the values of $t$ at which its velocity is zero is:
The length of the tangent drawn at the point \(P(1, 3\sqrt{3})\) on the curve is \(\frac{x^2}{3} + \frac{y^2}{27} = 4\):
The function \(f(x) = \frac{x}{2} + \frac{2}{x}\) has a local minimum at
The surface area of a cube is 150 sq. cm. If it is increased by 0.025 sq. cm, then the approximate increase in its volume (in c.c.) is:
If \(f(x)=ax^{3}+bx^{2}+cx+1\) attains an extreme value \(2\) at \(x=1\) and another extreme value at \(x=\frac{2}{3}\), then \(2b+3c\) is equal to
If a cylindrical tank of radius 3 m is filled with water at the rate of \(\frac{3}{2} \, m^3/sec \), then the rate of change of its water level in (m/sec) is:
Slope of the tangent to the curve \[ y=9x^2+7x^4+5 \] at the point \(x=1\) is
The length of the normal drawn to the curve \( 2x^{3}+2y^{3}=9xy \) at the point (2, 1) is
Approximate value of \( \sqrt[3]{345} \), when it is calculated with the application of derivatives, is
If the rate of increase in the surface area of a cube is 6 sq. cm./sec., then the rate of increase in its volume (in c. c./sec), when the length of its edge is 12 cm, is
If a sector of maximum area is made with a wire of length 40 cm, then the area (in sq cms) of that sector is
Let \(\mathbb{R}\) denote the set of all real numbers. Consider the polynomial function \[ f : \mathbb{R} \to \mathbb{R} \] defined by \[ f(x) = \frac{d^{10}}{dx^{10}}\left((x^2 - 1)^{10}\right), \qquad \text{for all } x \in \mathbb{R}. \] Here, \[ \frac{d^{10}}{dx^{10}}\left((x^2 - 1)^{10}\right) \] is the \(10^{\text{th}}\) order derivative of the function \((x^2 - 1)^{10}\). Then which of the following statements is (are) TRUE?
If the local maximum 'M' and local minimum 'm' of the function $f(x)=x-\frac{x^{2}}{2}-xe^{2-x}$ exist at $x=\alpha$ and $x=\beta$ respectively, then $2\alpha m+\beta M=$
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be such that $f(2+x)=f(2-x)\,\,\forall x\in\mathbb{R}$. If $f(x)$ is twice differentiable such that $f^{\prime}(1)=0$, then which one of the following is true?
If the length of the tangent at a point on the parabola $y^{2}=4ax$ is $4a\sqrt{5}$, then the length of the sub-normal at that point is
The minimum value of the function $f(x) = x^2 + \frac{250}{x}$ for $x > 0$ is:
The slope of the normal to the curve $y = 2x^2 + 3\sin x$ at $x = 0$ is:
A balloon, which always remains spherical, has a variable radius. The rate at which its volume is increasing with respect to its radius $r$ when $r = 5$ cm is:
If \[ y=(x-1)(x-2)(x-3)\cdots(x-100) \] and the value of \( \dfrac{dy}{dx} \) at \(x=0\) is equal to \[ \lambda\left(\frac{100!}{^{100}C_5}\right) \] then \( \lambda \) is:
Given \( f(x) = x - 1 \), \( h(1) = 4 \), and \( h'(1) = -2 \). If \( g(x) = (f[4(h(x)) + 3])^2 \), find the value of \( g'(1) \).
The point of inflexion for the curve \(y = (x - a)^n\), where \(n\) is an odd integer and \(n \ge 3\) is:
Let \(f:\mathbb{R}\to\mathbb{R}\) be defined by \[ f(x)=x^3-3x+1 \] Then the function \(f\) is:
In a Mahakumbh, a drone camera is moving along $3y = x^3 - 3$. When y-coordinate changes 9 times as fast as x-coordinate, it captures good quality pictures. Then one of the precise positions of the drone at that instant is